The mixed scalar curvature flow and harmonic foliations

We introduce and study the flow of metrics on a foliated Riemannian manifold $(M,g)$, whose velocity along the orthogonal distribution is proportional to the mixed scalar curvature, ${\text{Sc}}_{\mathrm{m}\mathrm{i}\mathrm{x}}$. The flow is used to examine the question: When a foliation admits a metric with a given property of $\Sc_{,\rm mix}$ (e.g., positive or negative)/? We observe that the flow preserves harmonicity of foliations and yields the Burgers type equation along the leaves for the mean curvature vector $H$ of orthogonal distribution. If $H$ is leaf-wise conservative, then its potential obeys the non-linear heat equation $\dt u=n\Delta_\calf,u +(n\beta_{\mathcal D}+\Phi),u+\Psi^\calf_1 u^{-1}-\Psi^\calf_2 u^{-3}$withaleaf-wiseconstant$\Phi$ and known functions ${\beta }_{\mathcal{D}}\ge 0$ and ${\mathrm{\Psi }}_i^{\text{calf}}\ge 0$. We study the asymptotic behavior of its solutions and prove that under certain conditions (in terms of spectral parameters of leaf-wise Schr"{o}dinger operator ${\mathcal{H}}_{\text{calf}}=-{\mathrm{\Delta }}_{\text{calf}}-{\beta }_{\mathcal{D}}\text{id}$) there exists a unique global solution $g_t$, whose ${\text{Sc}}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ converges exponentially as $t\to \mathrm{\infty }$ to a leaf-wise constant. The metrics are smooth on $M$ when all leaves are compact and have finite holonomy group. Hence, in certain cases, there exist $\mathcal{D}$-conformal to $g$ metrics, whose ${\text{Sc}}_{\mathrm{m}\mathrm{i}\mathrm{x}}$ is negative or positive.